Deterministic models in IE
Homework #4.2 Set B
Problem 3.3
Consider the following problem:
Maximize 1 + 32
subject to
1 22 0
21 + 2 4
51 + 32 15
1 , 2 0
a. Solve this problem graphically.
b. Solve the problem by the simplex method
Answer: a)
Graphical sol
4/28/2016
Problem 9.2 #2
Because of excessive pollution on the Momiss River, the state of
Momiss is going to build pollution control stations. Three sites (1, 2,
and 3) are under consideration. Momiss is interested in controlling the
pollution levels of t
540:399 Homework 1 Answers:
1. a)
b) This is a programming exercise. So there may be different ways of doing it. I will provide
one way using the state machine above. You will have to look closely at their programs to
determine whether or not they will wo
HW 7 Solution
#1.
(a) Problems 2 in Section 8.2
(b) As for the above problem, please formulate the above problem into the linear
programming
Sol)
(a)
1
8
1
1
2
1
2
4
2
4
4
5
3
Solved nodes
directly connected
to unsolved nodes
Closest connected
unsolved no
HW2 Solution
Exercise numbers are from the book W. Winston, Operations Research: Applications and
Algorithms, 4th edition
O Four Possible Solutions for the LP Problem (Section 3.3)
#1. Problems 3 in Section 3.3
Sol)
AB is x1 - x2 = 4. AC is x1 + 2x2 = 4.
HW1 Solution
O Review of Basic Linear Algebra
#1. Problems 1 in Section 2.2
Sol)
1 1
4
2 1 x1 = 6
1 3 x 2 8
#2. Problems 2 in Section 2.3
Sol)
1 1 1 4 1
1 2 0 6 0
1
1
1 4 1
1 2 0
0
1
2 2
1 2
This system has an infinite number of solutions of the f
1
Total Probability rule and Random Variables
2
Use the Total Probability Rule and random variables
An airport limo has 4 seats. Depending on demand, the number of reservations made has the following distribution: 5 6 r 3 4
f(r) .1 .2 .3 .4
From previous
1
Mean & Std Deviation of a Function of a Discrete random variable
2
Example: What is a function of a random variable?
r.v. X= number of defects on a part. Here is f(x):
x f(x) 0 1 .3 .5 2 .2
The rework cost is $2 per defect r.v. C = rework cost r.v. C(
1
Poisson Distribution
Poisson Distribution: Examples
2
r.v. X = number of events in an interval Possible values X=0, 1, 2, .
r.v. X = number of defects on a square inch of painted surface r.v. X = number of earthquakes in a year r.v. X = number of typo
1
Geometric Distribution
Geometric: Experiment and r.v.
2
Experiment
Conduct independent trials. The outcome on each trial is success with probability p and failure with probability 1-p. Stop at the first success.
R.v. X=number of trials to the first succ
1
Binomial Distribution
2
Binomial Distribution: the Situation
Experiment
There are N independent trials. The outcome for each trial is success with probability p and failure with probability 1-p.
Define r.v. X = number of successes in N independent trial
1
Mean and Standard Deviation of a Random Variable
2
Mean: Measure of Central Tendency
Guess the Mean
x 0 1 2 f(x) .15 .70 .15
Is this mean higher or lower
x 0 1 2 100 f(x) .14 .70 .14 .02
Expected Value of discrete r.v. X
3
Called:
Expected Value of X
1
Discrete Random Variables
Discrete Random Variable X
2
ex: flip a coin twice r.v. X = number of heads (now each possible outcome is attached to a number)
S = cfw_ HH , TH , HT , TT
2 1 1 0
Sample Space set of possible outcomes Assign a numerical value
1
Bayes Rule
2
Revisiting the Suppliers WITH A NEW QUESTION
A supplies 60% of parts with fraction defective 0.02 and B supplies 40% of parts with fraction defective 0.05. If I find a defective part, what is the probability it came from supplier A? Given:
1
Total Probability Rule and Independence
2
An old friend, the weighted average
Weighted Average: There are two suppliers A and B.
A supplies 80% of parts with fraction defective .01. B supplies 20% of parts with fraction defective .07. Greater than 0.
1
Conditional Probability and the Multiplication Rule
2
Women and Engineers at Rutgers
Experiment: randomly choose a Rutgers student
S=all students (sample space) Event E=student is engineer Event W=student is woman Rutgers is 50% women Engineers are 5%
1
Probability for experiments with discrete sample spaces
2
Count Number of Elements in the Sample Space
Discrete S: elements in S can be counted with the integers 1, 2, 3, .
finite: flip a coin S=cfw_H, T countably infinite: no. of calls in an hr S=cfw
1
Probability Basics: Sets
2
Experiment, Sample Space, Event
Curley brackets show a set cfw_
3
Experiment: Flip a coin twice
Curley brackets show a set
4
Experiment: components in parallel
Define sample space and its elements Show event that the system wo
Welcome to Engineering Probability 540:210 Professor Susan
What is probability?
Branch of math that deals with uncertainty Repeat operation again & again Each time, different outcome IE: How to operate efficiently under uncertainty
IEs Make Decisions Unde
ENGINEERING PROBABILITY Spring 2012 Homework #8 _ Solutions Topic Binomial Distribution
375. A binomial distribution is based on independent trials with two outcomes and a constant probability of success on each trial. a) reasonable b) independence assu
ENGINEERING PROBABILITY Spring 2012 Homework #7 _ Solutions Topic Discrete random variables & Probability distribution functions & Expected values
32. 34. 37. 315. 316. All probabilities are greater than or equal to zero and sum to one. a) P(X 1)=P(X=1)
ENGINEERING PROBABILITY Spring 2012 Homework #6 _ Solutions Topic More on Bayes Rule and Independence
2155. 2157. Let A = excellent surface finish; B = excellent length a) P(A) = 82/100 = 0.82 b) P(B) = 90/100 = 0.90 c) P(A') = 1 0.82 = 0.18 d) P(AB) =
ENGINEERING PROBABILITY Spring 2012 Homework #5 _ Solutions Topic Independence and Bayes Rule
2136. Let A denote the upper devices function. Let B denote the lower devices function. P(A) = (0.9)(0.8)(0.7) = 0.504 P(B) = (0.95)(0.95)(0.95) = 0.8574 P(AB)
Homework #4 Solutions Topic Total probability rule and independence
2109. Let R denote the event that a product exhibits surface roughness. Let N,A, and W denote the events that the blades are new, average, and worn, respectively. Then, 2111. a) (0.88)(